Numerical Analysis Seminars: Dimension reduction of dynamical systems
19 May 2016 14:00 in CM107
We start this general audience talk by defining some basic notions from dynamical systems. After that, a variety of numerical approaches for the visualization of 2D (un)stable manifolds are presented (lots of colorful pictures!). A special case is that of an inertial manifold. The restriction of a flow to such a manifold can reduce an infinite-dimensional system to a finite set of ordinary differential equations which captures all the long-time behavior, i.e. the so-called global attractor. Simple, yet effective approximations of inertial manifolds are discussed, as well as an algorithm to compute them accurately. Finally, these elements are unified in a foliation of phase space, which amounts to a pair manifolds at each point, some of which are invariant.
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